A mass M, initially moving at speed v, collides

and sticks to a mass m, initially at rest. Assume

M À m, and work in this approximation. What are

the ¯nal energies of the two masses, and how much

energy is lost to heat, in: (i) the lab frame?; (ii) The

frame in which M is initially at rest?

To analyze the collision between the two masses, we need to consider the principles of conservation of momentum and energy. In this scenario, mass M is moving with an initial speed v, while mass m is at rest. After the collision, the two masses stick together, forming a single combined mass. Let's break down the problem step by step, addressing both the lab frame and the frame where mass M is initially at rest.

1. Analyzing the Lab Frame

In the lab frame, we start with the following parameters:

• Mass M has an initial velocity v.
• Mass m is at rest (initial velocity = 0).

Conservation of Momentum

The total momentum before the collision can be expressed as:

Initial Momentum (p_initial) = M * v + m * 0 = Mv

After the collision, the two masses stick together and move with a common velocity V. The total momentum after the collision is:

Final Momentum (p_final) = (M + m) * V

By applying the conservation of momentum:

p_initial = p_final

Thus, we have:

Mv = (M + m) * V

From this, we can solve for the final velocity V:

V = (Mv) / (M + m)

Calculating Final Kinetic Energy

The initial kinetic energy (KE_initial) of mass M is:

KE_initial = (1/2) * M * v²

The final kinetic energy (KE_final) of the combined mass after the collision is:

KE_final = (1/2) * (M + m) * V²

Substituting V into the equation for KE_final:

KE_final = (1/2) * (M + m) * [(Mv) / (M + m)]²

KE_final = (1/2) * (M + m) * (M²v² / (M + m)²)

KE_final = (1/2) * (M * M²v²) / (M + m)

Energy Lost to Heat

The energy lost during the collision can be calculated as:

Energy Lost = KE_initial - KE_final

2. Analyzing the Frame Where M is Initially at Rest

Now, let’s shift our perspective to the frame where mass M is initially at rest. In this frame:

• Mass M has an initial velocity of 0.
• Mass m is moving with a velocity of -v (since we are in the frame of M).

Momentum Conservation in This Frame

The initial momentum in this frame is:

p_initial = M * 0 + m * (-v) = -mv

After the collision, the final momentum is:

p_final = (M + m) * V'

Applying conservation of momentum gives us:

-mv = (M + m) * V'

Solving for V':

V' = -mv / (M + m)

Final Kinetic Energy in This Frame

The initial kinetic energy of mass m is:

KE_initial = (1/2) * m * v²

The final kinetic energy of the combined mass is:

KE_final = (1/2) * (M + m) * (V')²

Substituting V' into the equation for KE_final:

KE_final = (1/2) * (M + m) * [(-mv / (M + m))]²

KE_final = (1/2) * (M + m) * (m²v² / (M + m)²)

KE_final = (1/2) * (m²v²) / (M + m)

Energy Loss in This Frame

Again, the energy lost during the collision can be calculated as:

Energy Lost = KE_initial - KE_final

In both frames, we can see that some kinetic energy is transformed into other forms of energy, primarily heat, due to the inelastic nature of the collision. The calculations in both frames yield the same energy loss, illustrating the principle of conservation of energy across different reference frames.